I'm looking for a news site for Mathematics which particularly covers recently solved mathematical problems together with the unsolved ones. Is there a good site MO users can suggest me or is my only bet just to google for them?

There is a book "Unsolved problems in number theory" (which I don't find particularly fitting my taste). But this isn't a site and it covers some parts of number theory only. In any case, a response to "Is there a good site?" is no, as we can see from several discussions on MO about the state of art in many areas/problems.
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Wadim ZudilinJun 28 '10 at 13:21

5 Answers
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As a counter-point to my somewhat flippant previous answer (which only really applies if one is a specialist in the field), if you are looking at a field in which you are not as much a specialist in, I suggest reading the articles from the Bulletin of the AMS. The articles are designed to be fairly up-to-date and expository in nature, and often gives the state of the art in their reviews.

Of course, a similar caveat as that to Helge's answer applies: the "news" maybe several months out of date. But considering the glacial paces at which a lot of mathematical refereeing takes place, I think it is quite okay.

Another suggestion Annals: to appear. Also other top journals. If a big problem gets solved, its solution probably gets submitted to a journal of this type, so its to appear lists are what you are looking for. Of course, you only learn about the solution of the problem a few years late (refereeing takes time), but you can be almost certain that the solution is actually correct.

There are many "bottom journals", for example, specialised ones, with better problems and nicer solutions. ("Top journals" indicate some kind of "mathematical fashion.")
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Wadim ZudilinJun 28 '10 at 13:13

@Wadim: I think your comment clearly illustrates that the original question needs clarification. Presumably every non-review/non-expository paper published solve some problem. The OP really ought to narrow down the question a bit.
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Willie WongJun 28 '10 at 13:32

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@Wadim: "big problem" is a "mathematical fashion" statement. Furthermore, there are plenty of reasons to care about "mathematical fashion", e.g. you want to get a job. But I'll stop here, since this line of discussion might get troublesome.
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HelgeJun 28 '10 at 13:56

@Willie: Every nonmathematical question is imperfect! There is no need for clarification, all possible responses can be argued. Especially if they are imperfect (no examples, proofs--nothing specially convincing).
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Wadim ZudilinJun 28 '10 at 14:00

Yes, Helge, I am more than familiar with the job problem. And, yes, I also have to stop here, as it is a dangerous theme.
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Wadim ZudilinJun 28 '10 at 14:03

Any paper worth reading should include some background material and a description of general progress in its introduction section. This is especially true of papers that actually solve a problem, rather than chipping away at some small technicality.

@Willie, but how to decide which paper is worth reading before looking inside? Even abstracts are hardly informative.
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Wadim ZudilinJun 28 '10 at 13:11

You don't expect somebody wanting to get an overview of recently solved problems in a subfield to wade through the enormous mass of papers that appear on arXiv every day, do you?
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Michal KotowskiJun 28 '10 at 13:11

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@Michal: yes. As for "enormous mass", maybe you're thinking of physics. In number theory (say), there are an average of perhaps 3-4 postings a day on the arXiv. For the subfields of number theory that I am especially interested in, that amounts to a few papers a week. P.S.: just because the papers appear every day doesn't mean you have to look every day...
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Pete L. ClarkJun 28 '10 at 13:17

@Michal: I read the listings for physics: hep-th, gr-qc, gen-ph, class-ph, hist-ph, soc-ph; and for maths: math.ap, math.ca, math.dg, math.fa, math-ph every day. Granted I do have a few years of experience in being able to quickly decide whether a paper is interesting or not (n.b. interesting $\neq$ quality).
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Willie WongJun 28 '10 at 13:20

@Wadim: a corollary of what I said is that "if you are not entrenched in the field, and if reading the introduction isn't helpful to your understanding of what's going on, then the paper is not going to be worth reading." I make no pretensions of being able to decide the "quality" of the reading by just the title and the abstract.
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Willie WongJun 28 '10 at 13:21